JANUARY 2012PAPERS

VOLUME 60

NUMBER 1

IETPAK

(ISSN 0018-926X)

Antennas Maximum Gain of a Lossy Antenna ...... ......... ........ ....... .. ......... ........ ......... . A. Arbabi and S. Safavi-Naeini . Experimental Validation of Performance Limits and Design Guidelines for Small Antennas .... ........ ......... ......... .. .. ........ ......... ........ D. F. Sievenpiper, D. C. Dawson, M. M. Jacob, T. Kanar, S. Kim, J. Long, and R. G. Quarfoth Substrate Integrated Waveguide (SIW) Leaky-Wave Antenna With Transverse Slots ... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ....... J. Liu, D. R. Jackson, and Y. Long Subwavelength Substrate-Integrated Fabry-Prot Cavity Antennas Using Articial Magnetic Conductor ....... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... Y. Sun, Z. N. Chen, Y. Zhang, H. Chen, and T. S. P. See A Recongurable Wideband and Multiband Antenna Using Dual-Patch Elements for Compact Wireless Devices ...... .. .. ........ ...... H. F. Abutarboush, R. Nilavalan, S. W. Cheung, K. M. Nasr, T. Peter, D. Budimir, and H. Al-Raweshidy Frequency-Recongurable Monopole Antennas .. ........ ......... ......... ........ ........ A. Tariq and H. Ghafouri-Shiraz Low Prole Fully Planar Folded Dipole Antenna on a High Impedance Surface ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ...... A. Vallecchi, J. R. De Luis, F. Capolino, and F. De Flaviis Crumpling of PIFA Textile Antenna ...... ......... ........ ......... ... ...... ........ ......... ......... .. Q. Bai and R. Langley . Higher Order Mode Excitation for High-Gain Broadside Radiation From Cylindrical Dielectric Resonator Antennas . .. .. ........ ......... ......... ........ ......... ......... ........ ......... .. D. Guha, A. Banerjee, C. Kumar, and Y. M. M. Antar On the Characteristics of the Highly Directive Resonant Cavity Antenna Having Metal Strip Grating Superstrate ...... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ...... A. Foroozesh and L. Shafai Nature of Cross-Polarized Radiations from Probe-Fed Circular Microstrip Antennas and Their Suppression Using Different Geometries of Defected Ground Structure (DGS) ... ......... ........ ......... ......... . C. Kumar and D. Guha . Single, Dual and Tri-Band-Notched Ultrawideband (UWB) Antennas Using Capacitively Loaded Loop (CLL) Resonators ..... ......... ........ ......... ....... .. ........ ......... ......... ........ . C.-C. Lin, P. Jin, and R. W. Ziolkowski . Leaky Wave Enhanced Feeds for Multibeam Reectors to be Used for Telecom Satellite Based Links ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ....... A. Neto, M. Ettorre, G. Gerini, and P. De Maagt Arrays 94 GHz Substrate Integrated Monopulse Antenna Array ......... ......... ........ ........ Y. J. Cheng, W. Hong, and K. Wu The Planar Ultrawideband Modular Antenna (PUMA) Array .... ......... ........ ....... S. S. Holland and M. N. Vouvakis A Two-Channel Time Modulated Linear Array With Adaptive Beamforming .. ......... ......... . Y. Tong and A. Tennant Aperiodic Array Layout Optimization by the Constraint Relaxation Approach . ......... ......... ........ ......... ......... .. .. ........ ......... ......... ...... T. N. Kaifas, D. G. Babas, G. S. Miaris, K. Siakavara, E. E. Vaadis, and J. N. Sahalos Time Reversal Based Broadband Synthesis Method for Arbitrarily Structured Beam-Steering Arrays . ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... D. Zhao, Y. Jin, B.-Z. Wang, and R. Zang A Generalized Hybrid Approach for the Synthesis of Uniform Amplitude Pencil Beam Ring-Arrays .. ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ... O. M. Bucci and D. Pinchera Polarimetry With Phased Array Antennas: Theoretical Framework and Denitions ..... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ....... K. F. Warnick, M. V. Ivashina, S. J. Wijnholds, and R. Maaskant An Amplifying Recongurable Reectarray Antenna ... ......... ......... . ....... ......... ..... K. K. Kishor and S. V. Hum Design of Retrodirective Antenna Arrays for Short-Range Wireless Power Transmission ..... .. .. Y. Li and V. Jandhyala .

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(Contents Continued on p. 1)

(Contents Continued from Front Cover) Anisotropic Impedance Surfaces for Linear to Circular Polarization Conversion ........ ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ..... E. Doumanis, G. Goussetis, J. L. Gmez-Tornero, R. Cahill, and V. Fusco Imaging and Propagation Transmitting-Mode Time Reversal Imaging Using MUSIC Algorithm for Surveillance in Wireless Sensor Network ... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... X.-F. Liu, B.-Z. Wang, and J. L.-W. Li UWB Microwave Imaging of Objects With Canonical Shape .... ......... ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ...... N. Ghavami, G. Tiberi, D. J. Edwards, and A. Monorchio Experimental Characterization of an UWB Propagation Channel in Underground Mines ........ ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... Y. Rissa, L. Talbi, and M. Ghaddar Statistical Prediction of Site Diversity Gainon Earth-Space Paths Based on RadarMeasurements in the UK .... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... C. Nagaraja and I. E. Otung Wideband Characterization of Backscatter Channels: Derivations and Theoretical Background ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ...... D. Arnitz, U. Muehlmann, and K. Witrisal Calibration of Electric Field Sensors Onboard the Resonance Satellite .. ........ ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ........ M. Sampl, W. Macher, C. Gruber, T. Oswald, H. O. Rucker, and M. Mogilevsky Numerical and Analytical Techniques A Second-Order Asymptotic Approximation for the Sommerfeld Half-Space Problem . ......... ..... ... ......... .. W. Lihh . Direct Rational Function Fitting Method for Accurate Evaluation of Sommerfeld Integrals in Stratied Media ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ...... T. N. Kaifas Integral Equation Modeling of Doubly Periodic Structures With an Efcient PMCHWT Formulation . ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ... S. Nosal, P. Soudais, and J.-J. Greffet Reduced-Order Models of Finite Element Approximations of Electromagnetic Devices Exhibiting Statistical Variability ...... ......... ........ ......... ......... ........ ......... ........ P. Sumant, H. Wu, A. Cangellaris, and N. Aluru Spherical ADI FDTD Method With Application to Propagation in the Earth Ionosphere Cavity ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ..... D. L. Paul and C. J. Railton Analysis of Directional Logging Tools in Anisotropic and Multieccentric Cylindrically-Layered Earth Formations .... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ...... G.-S. Liu, F. L. Teixeira, and G.-J. Zhang Analytic Transient Analysis of Radiation From Ellipsoidal Reector Antennas for Impulse-Radiating Antennas Applications ... ......... ........ ......... ......... ........ ......... ..... S.-C. Tuan, H.-T. Chou, K.-Y. Lu, and H.-H. Chou An Analytic Solution of Transient Scattering From Perfectly Conducting Ellipsoidal Surfaces Illuminated by an Electromagnetic Plane Wave .. ......... ......... ........ ......... ..... H.-T. Chou, S.-C. Tuan, K.-Y. Lu, and H.-H. Chou Greens Function Extraction for Interfaces With Impedance Boundary Conditions ...... ....... E. Slob and K. Wapenaar Electromagnetic Field of a Horizontal Innitely Long Wire Over the Dielectric-Coated Earth .. ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .. Y. J. Zhi, K. Li, and Y. T. Fang Decomposable Medium Conditions in Four-Dimensional Representation ...... I. V. Lindell, L. Bergamin, and A. FavaroCOMMUNICATIONS

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Novel UHF RFID Tag Antenna for Metallic Foil Packages ...... ......... ........ ......... .. J. Ryoo, J. Choo, and H. Choo Design of a Broadband All-Textile Slotted PIFA . ..... ... .. P. J. Soh, G. A. E. Vandenbosch, S. L. Ooi, and N. H. M. Rais . An Ultrawideband (UWB) Slotline Antenna Under Multiple-Mode Resonance ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ . X. D. Huang, C. H. Cheng, and L. Zhu A Compact Hepta-Band Loop-Inverted F Recongurable Antenna for Mobile Phone ... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... . Y. Li, Z. Zhang, J. Zheng, Z. Feng, and M. F. Iskander Hybrid Monopole-DRAs Using Hemispherical/Conical-Shaped Dielectric Ring Resonators: Improved Ultrawideband Designs ......... ......... ........ ......... ......... ........ ......... ......... ........ D. Guha, B. Gupta, and Y. M. M. Antar A Half Maxwell Fish-Eye Lens Antenna Based on Gradient-Index Metamaterials ...... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... Z. L. Mei, J. Bai, T. M. Niu, and T. J. Cui Dual-Polarized Planar Feed for Low-Prole Hemispherical Luneburg Lens Antennas .. ...... A. R. Weily and N. Nikolic TM Scattering by Perfectly Conducting Polygonal Cross-Section Cylinders: A New Surface Current Density Expansion Retaining up to the Second-Order Edge Behavior ..... ......... ......... .... G. Coluccini, M. Lucido, and G. Panariello . A Modication of the Kummers Method for Efcient Computation of the 2-D and 3-D Greens Functions for 1-D Periodic Structures ...... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .... S. P. Skobelev A Spatial Beam Splitter Consisting of a Near-Zero Refractive Index Medium .. ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... R.-B. Hwang, N.-C. Hsu, and C.-Y. Chin Extended Mode-Based Bandwidth Analysis for Asymmetric Near-Field Communication Systems .... Y. Tak and S. NamERRATA

377 379 385 389 393 398 402 407 412 417 421

Errata to Three-Dimensional Near-Field Microwave Holography Using Reected and Transmitted Signals . ......... .. .. ........ ......... ......... ........ ......... ......... ........ ... R. K. Amineh, M. Ravan, A. Khalatpour, and N. K. Nikolova List of Reviewers for 2011 ....... ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ......... .

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Digital Object Identier 10.1109/TAP.2011.2181917

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012

Maximum Gain of a Lossy AntennaAmir Arbabi, Student Member, IEEE, and Saeddin Safavi-Naeini, Member, IEEEAbstractAn upper bound on the achievable gain of a lossy antenna is derived. This limit depends on the antenna size and a newly dened loss merit factor, which is shown to be a measure of the antenna material loss. The derived limit extends the well-known upper limit on the ratio of antenna directivity to its quality factor to the lossy antennas. Optimal antenna current distribution for the maximum gain is found, and the radiation pattern and antenna efciency are also presented. Index TermsAntenna maximum gain, lossy antenna, quality factor, small antenna.

I. INTRODUCTION

A

N ANTENNA is an essential component of a wireless system. Small-size antennas with optimal gain and bandwidth are on high demand for compact power-efcient portable radios. However, there is a well-known tradeoff between the antenna gain, its size, and its achievable bandwidth. It is shown that arbitrary high gain can be achieved from an arbitrary small perfect electric conductor (PEC) sphere if the current distribution on the sphere is selected in a proper way [1]. Chu derived an upper limit on the antenna directivity and ratio of antenna directivity to quality factor for omnidirectional antennas with a nite number of vectorial spherical modes [2]. Indeed, regardless of size, there is no limit on the gain of a lossless antenna, but for achieving high gain, the current amplitude on the antenna surface should be large. Large current amplitude on the antenna generates large reactive stored energy around the antenna. This reactive energy increases the antenna quality factor. Following Chu, several researchers tried to quantify the tradeoff between the antenna directivity and the large reactive energy stored around the antenna [1], [3][6]. In [5] and [6], it is shown that there is a limit on the ratio of directivity to quality factor of a general lossless antenna. In all of these studies, the antenna is assumed to be lossless and its quality factor is regarded as a measure of the antenna input impedance bandwidth. Although a large quality factor represents high reactive energy and small bandwidth, the relation between the conventional quality factor and bandwidth becomes less useful for low and moderate values of quality factor [7]. In addition, low- antennas can be matched to a desired impedance over a bandwidth wider than what is determined by their quality factors by using lumped, quasilumped

or, as has recently been reported, by exploiting engineered materials as a part of the matching circuit [8]. Furthermore, in millimeter wave, Tetrahertz, and optical frequency antennas, radiation efciency reduction due to the antenna losses is more restrictive than the bandwidth. Based on these facts, the study of the material loss and optimizing the antenna current distribution in a way to minimize its material loss and maximize the gain allow us to optimize the antenna structure under more realistic conditions. To our knowledge, this investigation has not been performed yet. Although [9] considered the effect of energy loss due to surrounding medium of an antenna, the antenna itself was still assumed to be lossless and enclosed in a lossless sphere. Therefore, the antenna material loss that resulted from a high reactive near eld was not considered. In this paper, the effect of the antenna material loss on the maximum achievable gain is analyzed. To this end, the antenna is assumed to be enclosed in a virtual sphere and its elds outside the sphere are expanded in terms of orthogonal vectorial spherical modes whose coefcients are related to the current distribution inside the sphere. It is shown that each of the vectorial spherical modes has its own radiation efciency which depends on the antenna size and a newly dened material loss merit factor. It is observed that the mode efciencies, except for a few lowest order modes, are very small. This result is a manifestation of the nite rank of the free space Greens function whose more general form is discussed in [10]. II. PROBLEM DESCRIPTION It is assumed that the antenna radiates in free space and the smallest sphere that encloses the antenna has a radius of . The coordinate system is chosen in a way that the origin is the center of the sphere and its axis is oriented in the antennas maximum radiation direction. It is also assumed that the antenna is fed by a lumped source and, therefore, the direct radiation of the applied source is ignored. The electric and magnetic eld satisfy the Maxwells equations in the entire space (1a) (1b) where is the equivalent current density (2)

Manuscript received January 30, 2011; revised May 01, 2011; accepted July 02, 2011. Date of publication September 15, 2011; date of current version January 05, 2012. This work was supported by the National Science and Engineering Council (NSERC) of Canada and Research In Motion (RIM). The authors are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail: aarbabi@maxwell.uwaterloo.ca; safavi@maxwell.uwaterloo.ca). Digital Object Identier 10.1109/TAP.2011.2167934

is position dependent. It should be noticed and that the conductivity of the material can be considered as a part of . With this assumption, the current is nonzero only over the region occupied by the antenna, which is enclosed by the virtual sphere. Outside of this sphere, elds are solution to source free

0018-926X/$26.00 2011 IEEE

ARBABI AND SAFAVI-NAEINI: MAXIMUM GAIN OF A LOSSY ANTENNA

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Maxwells equations and can be expanded in terms of vectorial spherical harmonics [11]

and constitute a set of mutually noted that although orthogonal vectorial functions, they do not form a complete set as shown in (8a) and (8b) at the bottom of the page. The total radiated power by the antenna can be found using (3a) and (3b) with integration of the Poyntings vector (9) because of the orthogonality of the modes, the total radiated power is the superposition of the power radiated by each individual mode. Maximum radiation intensity, as was assumed at 0 direction and can be found from the beginning, is in the (3a) in the far-eld region as

(3a)

(3b) where order, is the spherical Hankel function of the second are the spherical harmonics (4) and is a differential operator dened as (5) and in (3a) and (3b) are coefcients of the The and modes, respectively. These coefcients can be found by projecting the volume current density on an orthogonal vectorial set. If represents the sphere volume, then (see Appendix A) (6a) (6b) and are given in (41a) and (41b) in Apwhere pendix A. It can be veried that and are mutually orthogonal (7a) (7b) (7c) where when is Kroneckers delta function which is nonzero only and, in this case, it is equal to , and are norms of and , respectively. These norms are given in (8a) and (8b) and are independent of . It should be

(10) Now the material loss will be considered. The dissipative loss of the antenna results from the antenna material loss and is given by

(11) It will be shown that for a given antenna size, the antenna gain only depends on a dimensionless parameter dened as (12) which is referred to as loss merit factor. Using this new parameter, material loss becomes (13) is an effective loss merit factor for the antenna and is dened according to (13). This parameter is equal to the loss merit factor of the antenna material if the antenna is made of only one type of material. For an inhomogeneous antenna, the effective loss merit factor is smaller than the largest merit factor of its different constituents.

(8a) (8b)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012

The volume current density can be written as the summation and of two parts: Its projection on the space spanned by and a part which is orthogonal to this space, that is (14) where is orthogonal to and (15) and and are projections of and are given by on normalized and

and by dening (20a) (20b) Equation (19) can be rewritten as

(21) (16a) (16b) and are efciencies of it can be seen from (21) that and modes. Finally, the antenna each individual gain can be found to be given by (22), shown at the bottom of the page. III. OPTIMIZATION OF THE GAIN In (22), the terms in the numerator and the denominator are all positive and is only present in the denominator. To maximize the fraction, the denominator can be minimized independently to zero. As was by setting the nonradiating current density does not radiate and according to mentioned in Section II, (18), it adds to the dissipated power; therefore, the optimum . Similar reasoning leads to antenna should have (23) By dening (24a) (24b) (18) (24c) (24d)

is the nonradiating part of the equivalent current density. The vanish identically at any point outside elds generated by the sphere volume . The nonradiating current density neinor modes outside the sphere. Further disther excites cussion on nonradiating current densities can be found in [12]. From (14) and (15), the norm of can be written as

(17) plugging the left-hand side of (17) in (13) gives

The antenna total input power is the summation of the radiated and dissipated power

Equation (22) can be rewritten as

(25)

(19)

and and Because of the symmetry of (25) with respect to and , equating partial derivatives of the with respect to

(22)

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to zero will result in similar equations and identical optimal values for these coefcients

(26a) (26b) Furthermore, the numerator is maximized by requiring (27) using (26) and (27), (25) is simplied as

(28)Fig. 1. Mode efciencies for TM ( of n for kR = 10 and M = 10 . ) and TE ( ) modes as a function

Finally,

and

give (29)

and (30a) (30b)

IV. RESULTS AND DISCUSSION Plugging in norms of and mode efciencies are found as from (8) into (20), the

(31a) (31b)

Fig. 2. Maximum antenna gain as a function of kR for different values of M for kR between 0.05 and 0.5. The dashed line curve shows Harringtons normal = (kR) + 2kR. gain G

These equations show that the mode efciencies and, therefore, and the antenna the maximum gain are only a function of size. For copper, silver, and gold, the loss merit factor ( ) at and for a dielectric microwave frequencies is on the order of 10 and a loss tangent of material with a permittivity of is about 5 . Fig. 1 shows the mode efciencies as a function of for 10 and . As can be seen and show step-like behavior and are from this gure, almost equal to each other for the same value of . This step-like behavior is reminiscent of the assumptions made by Chu and Harrington for omn-directional and directional antennas [1], [2]. For a directional antenna, based on the mode impedance for each of spherical modes, Harrington assumed that only modes

are propagating modes and other modes are in with cutoff. Our results show that the efciency of each individual spherical mode is independent of . The step-like dependency of efciencies on can be regarded as a cutoff, but the cutoff edge is not only a function of the antenna size but also antenna material (effective loss merit factor). For example, in Fig. 1, can be assumed to be propagating modes, modes with and other modes have small efciency and can be ignored. Figs. 2 and 3 show the maximum antenna gain for different . The dashed line curve shows Harringtons values of . It is clear normal gain dened as , the maximum gain that for a large and moderate value of is larger than what Harrington found using only modes with . This larger gain comes with higher quality factor and smaller input impedance bandwidth.

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Fig. 3. Maximum antenna gain as a function of kR for different values of M for kR between 0.5 and 5. The dashed line curve shows Harringtons normal kR kR. gain G

= ( ) +2

Fig. 5. Efciency of the antenna with maximum gain as a function of kR for different values of the antenna effective loss merit factor (M ).

V. CONCLUSION In this paper, a new fundamental limit on antenna gain was introduced. It was shown that although a lossless antenna regardless of its size can have arbitrarily high gain, the antenna dissipative loss limits the gain for a real antenna. The maximum gain is a function of the antenna size and its material loss merit factor. General plots for maximum gain for different antenna size and loss merit factor were provided. APPENDIX SPHERICAL HARMONICS COEFFICIENTS OF THE FIELD OF A VOLUME CURRENT DENSITY Coefcients of eld expansion in terms of vectorial spherical harmonics can be found from volume current and charge density as [11]

Fig. 4. Radiation pattern of the antenna with maximum gain with R and M .

= 10

2 =(33a)

Using (14), (16), and (30), the equivalent current distribution inside the sphere for an antenna with maximum gain is given by

(33b) where is the speed of light in vacuum and is the volume charge density related to current density by the continuity equation

(32) (34) Fig. 4 shows the radiation pattern of an antenna with the maximum gain for and . Since and are almost the same, the radiation patterns in different cutting planes containing the axis are similar. As can be seen from this gure, the side lobe level is about 20 dB. Fig. 5 shows the efciency of the antenna with maximum gain as a function of its size for different values of the effective loss merit factor. after substituting from (34) into (33a), the rst term in the right-hand side can be written as

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(42b) (35) REFERENCES the surface integral is zero because is assumed to be enclosed in the volume . The right-hand side of (33b) can be simplied as[1] R. F. Harrington, Time Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [2] L. J. Chu, Physical limitations of omni-directional antennas, J. Appl. Phys., vol. 19, pp. 11631175, Dec. 1948. [3] H. A. Wheeler, Small antennas, IEEE Trans. Antennas Propag., vol. AP-23, no. 4, pp. 462469, Jul. 1975. [4] R. C. Hansen, Fundamental limitations in antennas, Proc. IEEE, vol. 69, no. 2, pp. 170182, Feb. 1981. [5] R. L. Fante, Maximum possible gain for an arbitrary ideal antenna with specied quality factor, IEEE Trans. Antennas Propag., vol. AP-40, no. 12, pp. 15861588, Dec. 1992. [6] W. Geyi, Physical limitations of antenna, IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 21162123, Aug. 2003. [7] S. R. Best, The Foster reactance theorem and quality factor for antennas, IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 306309, 2004. [8] R. W. Ziolkowski and A. Erentok, Metamaterial-based efcient electrically small antennas, IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 21132130, Jul. 2006. [9] A. Karlsson, Physical limitations of antennas in a lossy medium, IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 20272033, Aug. 2004. [10] D. A. B. Miller, Fundamental limit for optical componenets, J. Opt. Soc. Amer. B, vol. 24, no. 10, Oct. 2007. [11] J. D. Jackson, Classical Electrodynamics, 3rd ed. New York: Wiley, 1999. [12] A. J. Devaney and E. Wolf, Radiating and nonradiating classical current distributions and the elds they generate, Phys. Rev. D, vol. 8, no. 4, pp. 10441047, 1973. Amir Arbabi (S06) was born in Malayer, Iran, in 1984. He received the B.Sc. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 2006, the M.Sc. degree in electrical engineering from the University of Waterloo, Waterloo, ON, Canada, in 2009, and is currently pursuing the Ph.D. degree in electrical engineering at the University of Illinois, Urbana-Champaign. He has been a Research Assistant with the Photonic Systems Laboratory, University of Illinois, Urbana-Champaign, since 2009. His research interests include plasmonic and nanophotonic devices, fast numerical methods for simulation and design of novel photonic devices, and fundamental features and limits in optics and electromagnetics. Mr. Arbabi was a recipient of the Ontario Graduate Scholarship and the Presidents Graduate Scholarship while at the University of Waterloo. He is a student member of the Optical Society of America and a reviewer for the IEEE PHOTONICS JOURNAL.

(36) and with similar reasoning, the surface integral has vanished. Using (35) and (36), (33a) and (33b) can be rewritten as

(37a) (37b) Using the following identities: (38) and (39) Equations (37a) and (37b) are further simplied as

(40a) (40b) and by deningSaeddin Safavi-Naeini (M00) was born in Gachsaran, Iran, in 1951. He received the B.Sc. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 1974 and the M.Sc. and Ph.D. degrees in electrical engineering from the University of Illinois, Urbana-Champaign, in 1975 and 1979, respectively. He joined the Department of Electrical and Computer Engineering, University of Tehran, as an Assistant Professor in 1980 and became an Associate Professor in 1988. Since 2002, he has been a Full Professor in the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada. His research interests and activities include numerical electromagnetics applied to radio-frequency/microwave/millimeter-wave systems and circuits, antennas and propagation, wireless communication systems, very-high-speed digital circuits, and optical communication systems. He has been a scientic and technical consultant to a number of national and international telecommunication industrial and research organizations since 1980.

(41a) (41b) equations (40a) and (40b) become (42a)

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012

Experimental Validation of Performance Limits and Design Guidelines for Small AntennasDaniel F. Sievenpiper, Fellow, IEEE, David C. Dawson, Member, IEEE, Minu M. Jacob, Student Member, IEEE, Tumay Kanar, Student Member, IEEE, Sanghoon Kim, Jiang Long, and Ryan G. Quarfoth, Student Member, IEEE

AbstractThe theoretical limit for small antenna performance that was derived decades ago by Wheeler and Chu governs design tradeoffs for size, bandwidth, and efciency. Theoretical guidelines have also been derived for other details of small antenna design such as permittivity, aspect ratio, and even the nature of the internal structure of the antenna. In this paper, we extract and analyze experimental performance data from a large body of published designs to establish several facts that have not previously been demonstrated: (1) The theoretical performance limit for size, bandwidth, and efciency are validated by all available experimental evidence. (2) Although derived for electrically small antennas, the same theoretical limit is also generally a good design rule for antennas that are not electrically small. (3) The theoretical predictions for the performance due to design factors such as permittivity, aspect ratio, and the internal structure of the antenna are also supported by the experimental evidence. The designs that have the highest performance are those that involve the lowest permittivity, have an aspect ratio close to unity, and for which the elds ll the minimum size enclosing sphere with the greatest uniformity. This work thus validates the established theoretical design guidelines. Index TermsBandwidth, dielectric resonator antenna, efciency, fractal, metamaterial, planar antenna, quality factor, slot antenna, small antenna.

I. INTRODUCTION MALL antennas have been an important topic of research for many decades, and interest in the eld is increasing with the development of new systems that require broadband antennas with a small form factor. The analysis of small antennas is generally considered to have begun with the work of Wheeler [1] and Chu [2], who established the theoretical limits that show how electrical size and bandwidth are related. Since this early work, numerous authors have revisited these theories, and have suggested further renements. Although slightly more accurate, all of these new theories share the same basic conclusions established in the 1940sthat size can only be reduced at the expense of bandwidth or efciency. Furthermore, the early

S

Manuscript received February 10, 2011; revised May 17, 2011; accepted July 12, 2011. Date of publication September 15, 2011; date of current version January 05, 2012. This work was supported by SPAWAR under Contract N66001-03-2-8938. D. F. Sievenpiper, M. M. Jacob, T. Kanar, S. Kim, J. Long, and R. G. Quarfoth are all with the University of California San Diego, La Jolla, CA 92093 USA (e-mail: dsievenpiper@eng.ucsd.edu). D. C. Dawson is with SPAWAR Systems Center Pacic, San Diego, CA 92152 USA (e-mail: david.c.dawson@navy.mil). Digital Object Identier 10.1109/TAP.2011.2167938

papers as well as others that followed have provided theoretical guidelines for other aspects of small antenna design. In general, the best performance will be achieved if the dielectric constant is as low as possible, if the aspect ratio is close to unity, and if the internal structure of the antenna is such that the elds ll the minimum size enclosing sphere with the greatest possible uniformity. Along with the work that has been done to develop theoretical limits, a large amount of effort has been put into developing specic antenna designs in an attempt to optimize the relationship between size and bandwidth. In the 64 years since Wheelers rst paper, thousands of new antenna designs have been published, and each year we continue to see new publications exploring every conceivable arrangement of metal shapes and dielectric regions. However, many of these designs have sub-optimum performance, and could have been predicted to perform poorly if the theoretical design guidelines were more clearly understood from the start. In addition, many antenna designs are proposed for which performance is overestimated, such as by ignoring losses or incorrectly calculating the true electrical size of the antenna. This challenges not only antenna engineers, who must address these unphysical performance claims, but also system engineers, who end up relying on performance metrics that are ultimately unachievable. These issues may be caused in part because the theoretical design guidelines are not widely understood, and in fact have never been rigorously validated experimentally. Unfortunately, it is impossible to experimentally prove a physical theoryit can only be disproven by contradictory experimental evidence. Nonetheless, a theory that has been tested extensively and found to be true in all tests is generally accepted as correct, at least until contradictory evidence is found. In the eld of small antennas, there have been many attempts to optimize antenna designs to get as close as possible to the theoretical limits. However, each of these antennas represents a local optimization, and in each case it is possible that the authors have simply not chosen the best design, and perhaps another one may be found that could exceed the theoretical limit. The purpose of this paper is to systematically extract experimental results from a sufciently large sample of existing designs to demonstrate that the Wheeler-Chu limit is valid and correct across a broad range of electrical sizes and bandwidths. We furthermore show that the design guidelines that have been established for permittivity, aspect ratio, and the internal structure of the antenna are also supported by the experimental evidence. This paper establishes several important facts: (1) When the size and efciency are correctly calculated, the measured band-

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SIEVENPIPER et al.: EXPERIMENTAL VALIDATION OF PERFORMANCE LIMITS AND DESIGN GUIDELINES FOR SMALL ANTENNAS

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width of an electrically small antenna does not exceed the theoretical limit, regardless of the design. (2) The theoretical limit for small antennas is also a good design guideline even for antennas that are not electrically small. (3) The experimental evidence supports the theoretical predictions that performance of a small antenna is maximized with low permittivity, low aspect ratio antennas, in which the elds ll the smallest enclosing sphere as uniformly as possible. For electrically small antennas, a class of wire cage designs appears to have a performance advantage compared to other types, while some of the new and popular concepts, such as fractals and metamaterials, do not appear to provide a performance advantage compared to conventional designs. There have been other studies comparing various specic antenna types to the theoretical limit, such as Best and Hannas recent paper in which they compared several different designs [3], and Bests paper involving specically planar designs [4]. However, to date there has not yet been a study which has systematically examined the body of experimental data to validate the theoretical limits over a wide variety of antenna types. Thus, the value of this paper is to demonstrate that the Wheeler-Chu limit has been extensively tested using 64 years of small antenna performance data, and has been found to be valid in all cases. The results shown here are also consistent with the design guidelines established decades ago [1]. It is expected that this will provide useful guidance for future small antenna designers. II. BACKGROUND ON SMALL ANTENNA THEORY In this section we give a brief overview of the theoretical analysis of small antennas and the results that are relevant to this study. For a more detailed examination of these theories, see for example the rst chapter in either of the books by Hansen [5] or Volakis [6]. The rst author to establish the link between antenna, bandwidth, and efciency was Wheeler [1]. He studied two simple small antennas, a cylindrical parallel plate capacitor and a cylindrical coil inductor. He calculated the radiation power factor for the capacitive antenna as (1) and for the inductive antenna as (2) where C or L is the capacitance or inductance, and G or R is the radiation shunt conductance or series resistance. He showed that the maximum power factor for a cylindrical antenna of either type, with circular area A and height b, is (3) , and denotes a shape factor that multiplies where the area A to obtain the effective area, as augmented by the eld outside the cylindrical volume. The shape factor approaches unity for thin, at capacitive antennas or long, thin inductors,

although it can be much larger for other shapes. For more details on the use of the shape factor, please refer to the original work [1]. Note that we have changed some variable names in order to be consistent throughout this paper. Wheeler also introduced an ideal spherical wire coil antenna [7], [8] that has a power factor of

(4) where a is the minimum radius of a sphere enclosing the antenna. Note that if the sphere is lled with a material having innite permeability, as Wheeler explains [9], expelling the avoidable stored energy from inside the antenna, then p can be increased by up to a factor of 3 compared to the air lled case. Its maximum value is fundamentally limited by the unavoidable stored energy outside the antenna, to (5) Wheeler also illustrated the relationship between power factor and bandwidth [9] which depends on the matching circuit and the allowable reection coefcient, as described by Fano [10]. We can recognize Wheelers denitions for the power factor in (1) and (2), as the inverse of the quality factor Q of an RC or RL circuit. By inverting (5) we nd a good approximation to the expressions for the minimum Q that are derived by other authors using more rigorous methods. In addition to this limitation on bandwidth, Wheeler also identied in his initial paper on this subject [1] several important guidelines for small antenna design which are relevant to the present study: (1) that the addition of an electrically large ground plane can potentially double p for a given volume, (2) that increasing the permittivity inside antenna decreases p roughly in proportion to , and (3) that increasing the permeability can increase p by up to a factor of 3. (4) He also later explained [9] that for non-spherical antennas, the power factor is reduced because the spherical volume is only partially utilized. Chu was the rst to derive the minimum quality factor Q of a small antenna based on an expansion of the eld in terms of spherical modes [2]. Q is generally dened [11], [12] by the ratio of stored energy W to radiated power P at a particular frequency for an otherwise lossless antenna, (6) where W is dened as (7) and and are the time-average, non-propagating, stored magnetic and electric energy. Unfortunately, Chu does not explicitly state a formula relating Q and size, thus requiring some further work by the reader to apply his results to antenna design. However, he does provide a plot showing a small dipole with an

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ideal matching circuit as described by Fano [10] that has a bandwidth approximately proportional to . Hansen [13] and later McLean [12] followed Chus analysis to derive an expression for the Q of the lowest order mode in terms of the antennas electrical size

Q through the maximum allowable voltage standing wave ratio VSWR, or s

(11) In most cases we are concerned with matching the rst one or two modes, however Villalobos [25] has also derived limits for matching higher order modes. Additional studies have focused on the limitations for specic types of antennas. Examples that are relevant to this study are as follows. Sten [26] studied antennas near a conducting plane and determined that the proper measure of antenna size is a sphere that encloses both the antenna and its image currents. Ida [27] studied dielectric loaded monopole antennas, showing that the efciency-bandwidth product is reduced for large values of permittivity. Thal studied spherical wire antennas [28], and loop antennas [29], and concluded that the Q values are at least 3 times the theoretical limits for TE mode antennas, or at least 1.5 times the theoretical limit for TM mode antennas. Gustafsson [30] examined various shapes and determined the theoretical limits on Q, showing that it increases for any shape that deviates signicantly from a sphere, and that the maximum performance appears at an aspect ratio falling in the range between 1 and 2. Ghorbani [31] studied microstrip antennas and concluded that although the addition of resonant structures within the microstrip pattern are usually added to increase bandwidth, they actually reduce the maximum bandwidth that could be achieved with an ideal matching circuit. This is consistent with the guidance given by Wheeler [9], because the elds associated with these resonant structures are conned to a subset of the overall antenna volume. Finally, Stuart et al. [32] studied multi-resonant antennas. The authors showed through example that although the Q given by (6) does not deviate from the fundamental limit, the Q implied by the impedance of the antenna, (12) can be signicantly different if the antenna has two closely spaced resonances. Furthermore, they found that neither quantity is a good predictor of the half-power VSWR bandwidth for multiresonant antennas. Although the Q of an electrically small antenna can never be lower than the limits of (9) and (10), the bandwidth given by (11) is an approximation which assumes a matched antenna, and it is most accurate when the bandwidth is dened in terms of a low VSWR. Furthermore, as Fano shows [10], maximizing the reection coefcient toward a given tolerance increases the available bandwidth within that tolerance. Thus, by designing the antenna to meet a sufciently high VSWR tolerance over the band of interest, it is possible to exceed the bandwidth predicted by (11). In summary, the equations for minimum Q are always correct, but the equation for bandwidth based on a given Q can be exceeded by making the antenna or matching circuit multi-resonant, and by maximizing the reection coefcient within that band. This is illustrated in Fig. 1.

(8) Hansen also showed that loss can be represented as an additional resistance in series with the radiation resistance, so Q can be reduced at the expense of efciency. Therefore a more useful quantity for comparing antenna performance is the quality factor divided by efciency, . Collin and Rothschild [11] approached the problem by subtracting the energy associated with radiation from the total energy to nd simple expressions for the Q of each mode. They give the value for the lowest order spherical mode as (9) McLean [12] derived the propagating and non-propagating elds, and calculated Q from the ratio of these terms, arriving at the same result as Collin, above. He also found that the Q for circularly polarized antennas involving both TM and TE modes together is (10) Although this result is often associated with circularly polarized antennas, Pozar [14] claried that this formula for Q is simply a result of using two modes, and is not specically a function of the polarization of the antenna. Other published papers have provided various other expressions for the radiation Q, including Fante [15], Geyi [16], Hansen and Collin [17], Thal [18], and Vandenbosch [19]. However, (9) and (10) above are generally accepted today as correct. Furthermore, the other variations that have been explored generally deviate from the expressions above by only a small amount. There has also been work to simultaneously optimize gain and Q, such as the preliminary work by Fante [20] which includes numerical results for maximum G/Q. This work was later contested by Thal [18]. Geyi [21] provides an analytical result, and nds that it is possible to simultaneously minimize Q to a value given in (10), while maximizing G/Q, giving a maximum gain of 1.5 for an omnidirectional antenna, or 3 for a directive antenna. Although most studies have focused on Q, the quantity that is of most interest to antenna engineers is frequency bandwidth, B. Among others, Geyi [22] addressed this issue, concluding that B and 1/Q are equivalent for antennas with , however, this claim has been disputed recently by Best [23]. In any case, when attempting to match a given load impedance there is a tradeoff between bandwidth and acceptable reection coefcient [10]. Yaghjian and Best [24] derived the relationship between B and

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Fig. 1. Example Smith chart plot for a single-resonant, matched antenna (solid line) consistent with (11) compared to a multi-resonant, unmatched antenna (dashed line). These curves illustrate that for a given reection coefcient toler, designing an antenna with multiple resonances and avoiding a perfect ance match can improve bandwidth, within the Bode limit.

This last discussion would seem to suggest that the fundamental limits on Q would have minimal utility for antenna design, because the quantity that system designers care about is bandwidth, not Q. However, multiple resonant modes in a single antenna must be orthogonal, either in polarization or space. For electrically small antennas, the two lowest order modes can be orthogonal in polarization, and can be designed to be close in frequency. This can indeed reduce Q and improve bandwidth, as illustrated by the fact that (10) for the case of two modes provides a Q that is one-half that of (9) for . However, higher spatial modes will generally occur at higher frequencies. If the structure is loaded with reactance elements to lower the frequency of the higher spatial modes, they will occupy a subset of the total antenna volume, thus further raising their Q. Finally, even if multiple tuned circuits are included in the matching network, the maximum bandwidth is ultimately governed by the Bode limit [33] (13) Lopez [34], [35] and Hansen [36], [37] have explained the potential bandwidth improvement for various numbers of tuned circuits. For example, for a VSWR of 2, one additional tuned circuit can improve the bandwidth by a factor of 2.3 and an innite number of tuned circuits would provide a theoretical bandwidth improvement of 3.8. However, in practice most of the benet is obtained with one or two tuned circuits, and an excessively complicated matching circuit would contain substantial losses. III. ANALYSIS PROCEDURE To compare various electrically small antenna designs to the theoretical performance limits described above, we extracted

experimental data from the published literature. A search for small and antenna using the online search engine IEEE Xplore at the end of 2010 yields 7484 papers, which is far too many to analyze. We limited our search to only publications in IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. This is based on the rationale that this is the premier journal for antennas, so any antenna design that has lasting impact would eventually be published here in some form. We also included the IRE Transactions, the predecessor to this journal, although there were no papers published in IRE Transactions that met the search criteria and that included sufcient data to quantify the antenna performance. This limit resulted in 763 papers, which is a reasonable number. Our approach obviously cannot nd every possible small antenna design, because there are many that only appear in conference papers or other journals. However, we needed to use consistent criteria for inclusion of papers, and this sample size is sufcient for us to draw meaningful conclusions. Upon examining each of the 763 papers, we determined that many could be eliminated based on title alone. Papers that discussed small reectors, small arrays, and other topics not related to electrically small antennas were excluded. Papers that focused on antennas embedded in materials other than free space, such as water, the human body, or other lossy media, were also excluded. Furthermore, papers that described ultrawideband (UWB) antennas were excluded, because designs aimed at multi-octave bandwidth generally are not electrically small, and usually involve different design approaches than electrically small antennas. For a similar reason, antennas focused on high frequency bands, such as millimeter wave antennas, or antennas integrated on a semiconductor substrate were excluded because those papers are generally focused on goals other than optimizing size or bandwidth. Nonetheless, published antennas that passed through our manual lter but were found to not be electrically small were still left in the data set because they provide some insight into the range of applicability of the fundamental limits as design guidelines. In addition to the criteria described above, multiband antennas were also excluded because they are typically designed to optimize the number and spacing of bands, rather than the width of a single band. Diversity antennas were excluded for a similar reason. Tunable antennas were excluded unless we could identify one tuning point as representative of that design. In general, if a paper did not include sufcient information to evaluate the antenna, such as frequency, bandwidth, size, efciency, gain, or plots from which this data can be extracted, then it was not included. Although active matching techniques such as non-Foster circuits can potentially exceed the limits discussed here, we considered only passive structures. Finally, since our goal was to validate the theoretical limit, we only included papers with measured data. By manually ltering the papers as described above, we obtained 112 published antenna designs that contained sufcient experimental data for us to analyze. For each paper, we recorded the center frequency and fractional bandwidth, or extracted these from the frequencies of the band edges, or from plots of . For papers that included efciency data, we used the values provided by the authors. For those that did not, we

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used radiation patterns and gain when these were available. We estimated the directivity using (14) where and are the 3 dB beamwidths in radians in the two orthogonal planes. Although this approximation is most correct for directive antennas, it is still sufcient for our needs here, where we aim to keep the errors to within a few tens of percent or less. We then estimated the efciency using the quoted gain and calculated directivity (15) We used this approach for 19 of the papers. For published articles that provided neither efciency nor a radiation pattern, but still quoted gain, we estimated the efciency by using (16) where the ideal gain depended on the type of antenna. Dipolelike antennas without a ground plane were assigned an ideal gain of 1.5, monopole-like antennas on a large ground plane that were vertically polarized with a null toward zenith were given an ideal gain of 3, and patch-like antennas on a large ground plane where there is one central lobe that rolls off rapidly toward the horizon were given an ideal gain of 6. We used this approach for 11 of the papers. For three of the papers involving moderately high Q designs on lossy dielectrics, the efciency was approximated using the measured antenna Q and the loss tangent of the dielectric,

Fig. 2. The method for determining the radius, a, of the smallest enclosing sphere. (a) For an antenna with no ground plane, it is the smallest sphere to enclose the entire antenna. (b) For antennas on a small ground plane with less radius, or closer than from an edge, the sphere encloses the entire than ground plane. (c) For antennas on an electrically large ground plane, the sphere encloses the antenna and its image currents.

(17) For these cases we calculated an implied Q from the bandwidth using (11). This approach was only used when efciency could not be estimated using any other means. For 39 of the papers, the efciency was either quoted as nearly 100% by the author, or was assumed to be nearly 100% based on the design. That assumption was only applied if there was no other data from which to extract efciency, and when such an assumption was considered reasonable, such as for low-Q antennas built using entirely low loss metals and dielectrics. While these methods are approximate, we expect that they will be accurate to within a few tens of percent or less, and errors of this magnitude will not have a signicant effect on the overall conclusions of this paper. The electrical size of the antenna was calculated as the radius of the smallest sphere which encloses the entire antenna, as shown in Fig. 2. For antennas that do not include a ground plane, this is straightforward. For those that include an electrically large ground plane, with a radius at the center of the operating band, then the sphere includes the antenna and the image currents, so the radius a equals the distance to the farthest

point on the antenna from the bottom center. For antennas on an electrically small ground plane, or closer than to the edge of the ground plane, the entire ground plane was included in the size. The value for k was taken at the center of the operating band. A maximum bandwidth efciency product, , was calculated using (9) for linearly polarized antennas, or (10) for circularly polarized antennas, and applying (11). We standardized all designs to a VSWR of 2, which is consistent with the requirements of many applications, and the vast majority of published papers. These equations were also applied as appropriate when the polarization was not stated or was indeterminate, but where a judgment of whether it involved one or two modes could be made from the symmetry of the antenna and the feed. In addition to recording size, bandwidth, and efciency data, we also analyzed the effects of design type, permittivity, and aspect ratio, to compare to the design guidelines discussed above. For antennas containing multiple dielectric materials, the permittivity of the material lling most of the resonant portion of the antenna was used. The aspect ratio was taken as the ratio of the largest to smallest dimension of the outside boundary of the antenna. For antennas on electrically large ground planes, the dimension normal to the ground plane was doubled in calculating the aspect ratio, so as to include the effect of the image currents, as shown in Fig. 2(c). We grouped the antennas into several categories depending on the style of design. Antennas such as patches, planar inverted F antennas (PIFAs) and other similar designs which included a ground plane and had a width greater than their height were designated as Planar [38][89]. Dipoles and other such structures were designated as Linear [90][105] as well as designs that involve dipole-like modes on metal sheets, regardless of their aspect ratio. This category also included any vertically polarized antenna on an electrically large ground plane that had a height

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Fig. 3. The measured product for 110 antenna designs published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION by the end of the year 2010. . Specic references on the outer edge of the performance limit are noted. The theoretical limits are derived by applying (11) to (9) and (10) using a VSWR of

exceeding its width, and therefore behaved as a monopole. A sub-category of the linear type was a class that we called Feed antennas [106][119]. These were antennas on small ground planes that were typically shaped as a mobile phone or other such object, in which a small resonant structure actually serves as an exciter or feed for a mode which involves the entire ground plane. This is a category for which under-reporting the true antenna electrical size by neglecting the ground plane is common. Dielectric resonator antennas (DRA) [120][123] also formed a separate category, as well as antennas that involved materials with relative permeability , which were designated as Magnetic [124][126]. One category that performed particularly well was called the Wire Cage [127][137] type. These antennas were generally complicated wire structures with roughly spherical shape, or having an aspect ratio close to 1. The nal categories included designs which take advantage of popular trends in antenna design, and attribute their performance to Fractal [138][141] or Metamaterial [142][147] features. However, antennas that only included a single period or unit of such concepts, such as a solitary split ring resonator, were generally lumped into one of the other categories as appropriate. For papers that included multiple designs, we chose the best performing design, or the smallest. There were two papers where the measured results are so far above the theoretical limits that they were labeled as Problematic [148], [149] and not included in the data set. In both of those cases, the issues with the results can be traced back to problems in how the measurements were performed. IV. RESULTS AND DISCUSSION The centerpiece of this work is shown in Fig. 3. We plot the bandwidth efciency product versus the electrical size for the 110 relevant small antenna papers published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and compare the measured results to the theoretical limits. The curves

representing the theoretical limits are derived by applying (11) to (9) and (10) using a VSWR of , and including efciency to obtain

(18) where for linearly polarized or single-mode antennas, and for circularly polarized or dual-mode antennas. Note that (9) and (10) for the minimum Q are inviolable. However, direct data for Q is not readily available for most published antennas, so we are using measured bandwidth as a proxy for Q through (11). This is based on the assumption of a self-matched antenna without additional matching circuits, which is consistent with the vast majority of published antennas. It is possible to exceed this bandwidth-Q relationship of (11) by using additional matching circuits. However, it is never possible to exceed the Bode limit of (13) for a given Q [33]. Thus, although the Friedman antenna [48] stands out as exceeding the theoretical limits curves plotted in Fig. 3, it does not actually exceed the Wheeler-Chu limit. Furthermore, the potential benets of each additional matching circuit are well established [34][37]. The double-tuned matching circuit of the Friedman antenna can be expected to provide a bandwidth improvement factor of up to 2.8, [34][37] and the measured product of this antenna is well within this limit. It should also be noted that the Friedman antenna is similar in concept to the widely cited Goubau antenna [150]. The two are identical in electrical size and bandwidth performance. The exceptional performance of the Goubau antenna is often credited to its multi-mode design. In fact, both antennas include complicated matching networks. While Friedman explicitly uses an effective circuit approach to design his antenna, Goubau achieves the same result with wire loops, slots, and other features integrated throughout his antenna.

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 1, JANUARY 2012

There are several important observations to note in Fig. 3. (1) No electrically small antennas have been published in this journal that exceed the theoretical limit. (2) That limit also provides a good guideline for the maximum bandwidth even for antennas of moderate to large electrical size. (3) For electrically small antennas, wire cage designs appear to have a performance advantage compared to other types. (4) For moderate electrical size there are many standard planar or linear designs that can come close to the theoretical limit. (5) Dielectric resonator antennas perform relatively poorly because they are based on high dielectric materials, as predicted by the design guidelines discussed above. (6) Magnetic antennas are expected to have a performance advantage of up to a factor of three compared to other types. However very few antennas based on magnetic materials appeared in our sample of papers. There is one design that does achieve good performance, so this may be a promising area for future research if low-loss magnetic materials can be realized at frequencies of interest. (7) Fractal and metamaterial designs, although popular in recent years, do not appear to provide any performance advantage compared to more conventional antenna types. From these observations, we can draw several important conclusions about the relative merits of different antenna types, and the relationship between these performance variations and the established theoretical design guidelines. In general, the high-performance wire cage designs include low permittivity, low aspect ratio, and they have their elds evenly distributed throughout their volume or surface, so they are consistent with established design guidelines discussed above. Furthermore, the dielectric resonator designs involve high permittivity materials, so their poor bandwidth performance relative to their size is expected. Finally, the poor performance of metamaterial and fractal designs is consistent with the idea that the elds associated with the antenna should ll the smallest enclosing sphere as uniformly as possible. These designs typically involve highly resonant structures embedded within the antenna which tend to concentrate the elds to those regions, so they effectively use a subset of the available volume. In addition to examining the relative performance of different antenna classes, we also specically examined the effect of permittivity and shape. Fig. 4 shows the measured product divided by the theoretical limit given by (18) for the ka value of each antenna as a function of permittivity. For antennas that include multiple materials, the permittivity corresponds to the material lling the main resonant structure of the antenna. Note that the maximum measured performance decreases with increasing dielectric constant over a wide range of antenna designs. Wheelers original paper [1] instructed that for a capacitive antenna with a shape factor , the power factor is reduced by (19) Thus, for a shape factor of unity, corresponding to a thin, at capacitor, the performance is reduced approximately by the inverse of the permittivity. This performance roll-off is less severe for higher shape factors, so a roll-off is not a strict rule, but

Fig. 4. The measured product divided by the theoretical limit, compared to the relative permittivity of the material lling the antenna. The performance , is reduced with increasing permittivity. The dashed line shows a trend of an approximation from Wheelers paper, for a shape factor of 1. Most of the designs that lie above the dashed line actually contained multiple materials, and we have recorded the highest permittivity value, thus overestimating the effective permittivity in these cases.

it is still a good overall design guideline for many types of antennas. Also, from the spread of points in Fig. 4 it is obviously possible to achieve even worse performance, such as by inefciently using the antenna volume. Although an estimate of the shape factor for each antenna would allow us to more directly compare the data with Wheelers formula, we must remember that these antennas include a wide variety of internal structures, and many deviate signicantly from an ideal small capacitor. Thus, the concept of the shape factor would be difcult to apply directly to this entire data set. It is worth noting that many of the dielectric resonator antennas lie very close to Wheelers prediction. There are also several antennas that lie above the theoretical curve, and in most of these cases the discrepancy from the theory has to do with the choice of permittivity. These designs either contain multiple dielectric materials [66], [72][74], [89], or the elds extend partially into air regions within or around the antenna [85], [103]. In these cases we have recorded the highest permittivity among several materials that make up the resonant portion of the antenna, and therefore have clearly overestimated the effective permittivity. Accurately determining the true effective permittivity would be difcult, so these data points are left as exceptions. There are two additional cases that either involve magnetic materials [124] which would be expected to have higher performance, or complex matching circuits [48] which have been discussed above. We can observe that as a general rule, the maximum measured performance is reduced with increasing permittivity, and that assuming a performance reduction of roughly as a useful guideline for many designs, particularly if the effective permittivity can be accurately calculated. The maximum performance also decreases with increasing aspect ratio, as shown in Fig. 5. For this plot, we have used the ratio of the largest to smallest exterior dimensions of the antenna, regardless of orientation, so our aspect ratio is always greater than one. For antennas that involved an electrically large

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Fig. 5. The measured product divided by the theoretical limit, compared to the aspect ratio of the antenna. The performance is reduced with increasing aspect ratio. The dashed line shows a good linear approximation to Gustaffsons , curve for vertically polarized cylindrical antennas with roughly corresponding to some of the antennas designated here as the Planar , corresponding approxitype. The dotted curve is for mately to our Linear type.

ground plane, the image currents were also included in these dimensions. Gustaffson [30], calculated the performance reduction versus aspect ratio for various ideal shapes. He showed that for most shapes the maximum performance is achieved with an aspect ratio between 1 and 2, and the performance diminished with aspect ratio at various rates depending on the antenna shape. Note that his denitions for antenna shapes do not correspond exactly to our categories. Nonetheless, some comparisons can still be made. In Fig. 5 we have added theoretical curves which are linear approximations to Gustaffsons curve for vertically antennas having a cylindrical shape. For cylindrical antennas with a , this curve can be compared with some of the antennas in our Planar category. The theoretical performance decreases by approximately a factor of 10 for each decade increase in the diameter/height ratio. In other words, the performance is inversely proportional to the aspect ratio. Not all of these designs have a circular cross section [76], so it is difcult to compare the aspect ratio directly with Gustaffsons ideal curves. A separate line is included for cylindrical antennas with , which describes some of the antennas in our Linear category. A well-designed example is Noguchis [101] dual mode helix antenna. Although a perfect match to ideal antenna types is not to be expected for so many different designs, the trends in measured antenna performance are still generally consistent with the theoretical predictions. For many designs, and particularly for planar type antennas, it is a useful guideline to assume that performance may be reduced at least in proportion to the inverse of the aspect ratio. V. CONCLUSION We have demonstrated that the Wheeler-Chu limit for electrically small antenna performance is supported by experimental evidence for all papers containing measured data that have been published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. We argue that this is a sufciently large data set to validate the theoretical limit. We have also shown that

the limit serves as a good design guideline even for antennas that are not electrically small. We have further shown that the design guidelines for performance reduction with permittivity and aspect ratio agree with the theoretical predictions, to the extent that such a comparison can be made for a wide variety of antenna types. The expected performance for most antennas degrades approximately with the inverse of permittivity, and for planar type antennas it degrades approximately with the inverse of aspect ratio. Finally, we have shown that antenna types in which the resonant structure is restricted to a subset of the overall antenna volume perform poorly relative to those in which the elds are evenly distributed within the minimum size enclosing sphere. All of these conclusions are consistent with the existing theoretical design guidelines. In general, to design an electrically small antenna with the largest possible bandwidth efciency product, the antenna should have a low permittivity, an aspect ratio close to unity, and should have the stored elds distributed evenly throughout its volume. For this reason, in the electrically small regime, wire cage designs appear to have an advantage compared to other types. However, at larger scales there are many other designs that perform well compared to the theoretical limits. ACKNOWLEDGMENT The authors would like to thank S. Best for very helpful discussions about the relationship between Q and bandwidth, and multiresonant antennas. REFERENCES[1] H. A. Wheeler, Fundamental limitations of small antennas, in Proc. IRE, 1947, vol. 35, pp. 14791484. [2] L. J. Chu, Physical limitations of omni-directional antennas, J. Appl. Phys., vol. 19, pp. 11631175, 1948. [3] S. R. Best and D. L. Hanna, A performance comparison of fundamental small-antenna designs, IEEE Antennas Propag. Mag., vol. 52, pp. 4770, 2010. [4] S. Best, Optimization of the bandwidth of electrically small planar antennas, presented at the Antenna Applications Symp., Monticello, IL, 2009. [5] R. C. Hansen, Electrically Small, Superdirective, And Superconducting Antennas. Hoboken, NJ: Wiley-Interscience, 2006. [6] J. L. Volakis, C.-C. Chen, and K. Fujimoto, Small Antennas: Miniaturization Techniques and Applications. New York: McGraw-Hill, 2010. [7] H. A. Wheeler, The spherical coil as an inductor, shield, or antenna, in Proc. IRE, 1958, vol. 46, pp. 15951602. [8] H. A. Wheeler, The radiansphere around a small antenna, in Proc. IRE, 1959, vol. 47, pp. 13251331. [9] H. Wheeler, Small antennas, IEEE Trans. Antennas Propag., vol. 23, pp. 462469, 1975. [10] R. M. Fano, Theoretical limitations on the broadband matching of arbitrary impedances R.L.E. Tech. Rep. 41, Jan. 2, 1948. [11] R. Collin and S. Rothschild, Evaluation of antenna Q, IEEE Trans. Antennas Propag., vol. 12, pp. 2327, 1964. [12] J. S. McLean, A re-examination of the fundamental limits on the radiation Q of electrically small antennas, IEEE Trans. Antennas Propag., vol. 44, p. 672, 1996. [13] R. C. Hansen, Fundamental limitations in antennas, Proc. IEEE, vol. 69, pp. 170182, 1981. [14] D. M. Pozar, New results for minimum Q, maximum gain, and polarization properties of electrically small arbitrary antennas, in Proc. EuCAP 3rd Eur. Conf. on Antennas and Propagation, 2009, pp. 19931996. [15] R. Fante, Quality factor of general ideal antennas, IEEE Trans. Antennas Propag., vol. 17, pp. 151155, 1969. [16] W. Geyi, A method for the evaluation of small antenna Q, IEEE Trans. Antennas Propag., vol. 51, pp. 21242129, 2003.

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